Problem: Is ${743052}$ divisible by $3$ ?
A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {743052}= &&{7}\cdot100000+ \\&&{4}\cdot10000+ \\&&{3}\cdot1000+ \\&&{0}\cdot100+ \\&&{5}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {743052}= &&{7}(99999+1)+ \\&&{4}(9999+1)+ \\&&{3}(999+1)+ \\&&{0}(99+1)+ \\&&{5}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {743052}= &&\gray{7\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{3\cdot999}+ \\&&\gray{0\cdot99}+ \\&&\gray{5\cdot9}+ \\&& {7}+{4}+{3}+{0}+{5}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${743052}$ is divisible by $3$ if ${ 7}+{4}+{3}+{0}+{5}+{2}$ is divisible by $3$ Add the digits of ${743052}$ $ {7}+{4}+{3}+{0}+{5}+{2} = {21} $ If ${21}$ is divisible by $3$ , then ${743052}$ must also be divisible by $3$ ${21}$ is divisible by $3$, therefore ${743052}$ must also be divisible by $3$.